LOW-COMMUNICATION FFT WITH FAST MULTIPOLE METHOD Cris Cecka, Senior - - PowerPoint PPT Presentation

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LOW-COMMUNICATION FFT WITH FAST MULTIPOLE METHOD Cris Cecka, Senior - - PowerPoint PPT Presentation

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May 8-11, 2017 | Silicon Valley LOW-COMMUNICATION FFT WITH FAST MULTIPOLE METHOD Cris Cecka, Senior Research Scientist. May 11, 2017 THE FAST FOURIER TRANSFORM Operation Count: 4 N log 2 N 6 N + 8 2 SPLIT-RADIX FFT Algorithm 3

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May 8-11, 2017 | Silicon Valley

Cris Cecka, Senior Research Scientist. May 11, 2017

LOW-COMMUNICATION FFT WITH FAST MULTIPOLE METHOD

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THE FAST FOURIER TRANSFORM

Operation Count: 4N log2 N − 6N + 8

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3

SPLIT-RADIX FFT

Algorithm

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SPLIT-RADIX FFT

Profile

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FMM-FFT

Edelman et al. 1999

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STRUCTURED DENSE MATRICES AND FMM

  • SVD:
  • Low-Rank:
  • Hierarchically LR:
  • H-Semi-Separable:
  • H2-Matrix/FMM

A = U D V ∗

K = U ˜ Kr×r V ∗ KIJ = UI ˜ KIJ V ∗

J

KIJ = UI ˜ U˜

I ˜

K˜

I ˜ J ˜

V ∗

˜ J V ∗ J

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7

FMM-FFT

Algorithm

MM,P = diag(IM, C1, . . . , CP −1) [Cp]mn = ρp h cot ⇣ π M ⇣ n − m + p P ⌘⌘ + ı i

} 2D M × P FFT

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COT FMM

  • One dimensional
  • Uniform — integers are source/target
  • Periodic
  • Distributed
  • Size M-by-M
  • P of them!
  • Interleaved

[Cp]mn = ρp h cot ⇣ π M ⇣ n − m + p P ⌘⌘ + ı i

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FMM OPERATORS

Each operator is an (implicit) matrix.

M/2L Q Q Q S2M M2M M2M M2L M2L L2L L2L L2L L2T L2T S2T

  • S: “Source”
  • T: “Target”
  • M: “Multipole”
  • L: “Local”

S2T M2L B=2 3 L=4

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PARAMETERS OF THE FMM-FFT

  • FFT
  • FMM
  • Rank
  • Base level
  • Leaf box size
  • Leaf level

N = M P Q B

ML

L = log2(M/ML) (N, P, ML, Q, B)

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DISTRIBUTED FMM

All2All Gather All2All Gather Halo 2b Halo 2b Halo 1b Halo 2b Halo 2b Halo 1b

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INTERPOLATIVE FMM

  • Same operators across all boxes
  • Same operators across all levels
  • Almost same operators across all FMMs

zj = cos ✓(2j + 1)π 2Q ◆ `i(z) = Y

0k<Q k6=i

z − zk zi − zk

S2M M2M M2L L2L L2T

Cij = `m(tI

i ) `q(z ˜ I m) C(z ˜ I q , z ˜ J r ) `r(z ˜ J n ) `n(sJ i )

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TENSOR REPRESENTATIONS

  • Input:
  • Output:

Aijk` := A[i + j ∗ ldA<1> + k ∗ ldA<2> + ` ∗ ldA<3>],

Sn ≡ Spm ≡ Spmb Tn ≡ Tpm ≡ Tpmb

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S2M/L2T

S2Mqm = `q(sm)

sm = −1 + 2m + 1 ML

Computed with single BatchedGEMM

ML

(p−1)qb = S2Mqm Spmb

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BATCHED MATRIX-MATRIX MULTIPLY

cublas<T>gemmStridedBatched in cuBLAS 8.0

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S2M/L2T

Tpmb = L2Tmq Lpqb = ⇒ Tpm[b] = Lpq[b] S2Mqm

Mpqb = S2Mqm Spmb = ⇒ Mpq[b] = Spm[b] S2M T

qm

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M2M/L2L

M2M ±

qk = `q

✓zk ± 1 2 ◆

M`

pqb = M2Mqk M`+1 pk(2b)

Computed with single BatchedGEMM

L`+1

pq(2b) = L2Lqk L` pkb + L`+1 pq(2b)

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S2T/M2L

  • Also Level-3 Linear Algebra computations, but no BLAS primitives.
  • CUSTOM KERNELS

Tpib = S2Tp(j−i) Spjb

S2Tpk = ( cot π

N (p + Pk)

  • p > 0

δk0 p = 0

L`

pib = M2L` pijs M` pj(b+s)

M2L`

pijs = cot

⇣ π 2` (zj 2 − zi 2 + s) + π N (p + 1) ⌘

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INTERPOLATIVE FMM

P(4ML-1) QML QML 2Q2 2Q2 4(L-B)PQ2

Storage Operator Compute

2PMQ 2PMQ 3P2LML2 4(2L-2B)PQ2 4(2L-2B)PQ2 3(2L-2B)PQ2

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ALGORITHM

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PROFILE

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FMM-FFT PROFILE

S2M M2M Halo S2T M2L

}

L2L L2T

2D FFT

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2xK40c FMM-FFT

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2xP100 FMM-FFT

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8xP100 FMM-FFT

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FMM BREAKDOWN

  • T=ComplexDouble, A=2xP100
  • B-GEMM and S2T dominate
  • Small N
  • Latency — Use 1 Level
  • Large N
  • Compute

Components

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EFFICIENCY

  • >95% BatchedGEMM
  • 60% S2T/M2L
  • >90% FMM-FFT
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PARAMETER DEPENDENCE — ML

  • Trade #levels for S2T comp
  • Flop count not enough
  • Increase the intensity
  • Tune performance for ML=64
  • T=Z, A=2xP100, N=227, P=256, B=3, Q=16

Points per box per FMM

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PARAMETER DEPENDENCE — P

  • Flops/Intensity approx constant
  • Trade #levels for #FMMs
  • Large P good
  • Fill up B-GEMM
  • More square 2D FFT
  • T=Z, A=2xP100, N=227, ML=64, B=3, Q=16

Number of FMMs

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PARAMETER DEPENDENCE — B

  • Not very significant
  • Scale to 128 GPUs w/o complications
  • T=Z, A=2xP100, N=227, P=256, ML=64, Q=16

Base Level

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PARAMETER DEPENDENCE — Q

  • Weak performance dependence
  • Accuracy tuning
  • T=Z, A=2xP100, N=227, P=256, ML=64, B=3

Quadrature Order

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FUTURE

  • Integration into CUFFT
  • Application to 2D/3D FFTs?
  • Convolutions
  • NUFFT

, Sparse FFT

  • Volta predictions and measurements
  • Mixed precision (e.g. FP16 far-field) to use Tensor Core?
  • Persistent Matrix Batched GEMM (cuBLAS optimization)
  • Staged Persistent Matrix Batched GEMM (cooperative groups, RNNs)
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CONCLUSION

  • FMM-FFT trades 2/3 communication in 1D FFT for P FMMs
  • Viable on highest comp:comm architecture available
  • Detailed implementation that relies heavily on existing primitives
  • Primitives >95% efficient
  • Two custom dense kernels >60% efficient
  • Entire FMM-FFT >90% efficient
  • Tunable accuracy-performance tradeoff
  • Compute model accurately predicts performance
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May 8-11, 2017 | Silicon Valley

THANK YOU